While it is natural to posit that the distance between brain states is an important constraint on the ease of a state transition, there are other important principles that are also likely to play a critical role. Paramount among them is the architecture of the network available for the transmission of control signals. We therefore now turn to the question of which features of the network predict the amount of energy required for each transition from the pre-stimulation state to a good memory state. To address this question, we considered the empirical networks as well as the topological and spatial null model networks discussed earlier. We find that the optimal control input energy required for these state transitions differs across network types (one-way ANOVA \(F(2,75) = 4.00\) , \(p = 0.03\) ). In post-hoc testing, we found that the optimal control energy was significantly different between the empirical network and the topological null network (two-tailed \(t\) -test: \(t = -2.6\) , \(p = 0.01\) ) (Fig. REF A), but not between the empirical network and the spatial null network (\(p>0.05\) ). This observation suggests that the spatial embedding that characterizes both the real network and the spatial null network may increase the difficulty of control. In supplemental analyses, we test two additional spatially embedded null models that further preserve degree distribution and strength sequence, and we find similar average energies to the empirical and spatial null models discussed here (see Supplement). We hypothesized that the difference in optimal control energy could be mechanistically explained by the determinant ratio, a recently proposed metric quantifying the trade-off between connection strength (facilitating control) and connection homogeneity (hampering control) [1]}. Intuitively, a network with a high determinant ratio will have weak, homogenous connections between the control nodes and nodes being controlled. We found that across all networks the determinant ratio explains a significant amount of variance in energy after accounting for network type (linear mixed effects model with network type and determinant ratio as fixed effects: \(\chi ^{2} = 12.2\) , \(p = 4.7\times 10^{-4}\) ) (Fig. REF B). These results support the notion that spatial embedding could impose energy barriers by compromising the trade-off between the strength and homogeneity of connections emanating from the stimulating electrode.
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